Global regularity of 2D Navier–Stokes free boundary with small viscosity contrast

This paper studies the dynamics of two incompressible immiscible fluids in 2D modeled by inhomogeneous Navier-Stokes equations. We prove that if initially viscosity contrast is small then there global-in-time regularity. result has been proved recently [32] for $H^{5/2}$ Sobolev regularity interface. Here we provide a new approach which allows to obtain preservation natural $C^{1+\gamma}$ H\older interface all $0<\gamma<1$. Our proof direct and low initial velocity without any extra technicality. It uses quantitative harmonic analysis bounds $C^{\gamma}$ norms even singular integral operators on characteristic functions domains [21].